The Second Extreme Gas Dynamic Activity P M V Subbarao Professor Mechanical Engineering Department The devil caught by Sir Ernst Mach’!!!

Two Extreme Flyers Flying of point or zero degree conical object at high speeds : Leads to derivation Speed of Sound. Flying of Large/180 degree conical object at high speeds : generation of A discontinuity.

Weak Shock Due to Hypersonic Sleek flyers One does find in some of the ancient writings of India some descriptions of advanced scientific thinking which seemed anachronistic to the age from which they come." - Charles Berlitz - Mysteries from Forgotten Worlds.

High Angle Flying Objects Sleek Bodies at supersonic Speeds Bluff Bodies at supersonic Speeds The second Extreme of Gad Dynamic Actions

Prandtl Theory The first theories about supersonic shock waves and flow came from Prandtl and Theodore Meyer, one of his students, in 1908. Four decades before the first supersonic aircraft. He didn’t continue working on this theory till the 1920s. Adolf Buseman worked with Prandtl to developed a method for designing a supersonic nozzle in 1929. Ludwig Prandtl

Prandtl – Meyer Analysis of A Shock A control volume for this analysis is shown, and the gas flows from left to right. The control volume is having a very narrow width. No chemical reactions. There is no friction or heat loss across this CV. No work transfer is possible as this is so narrow. The increase of the entropy is fundamental to any thing that happens in this world.

Conservation of Mass Applied to 1 D Steady Flow The area of the disturbance is not varying in flow direction. Conservation of momentum: The momentum is the quantity that remains constant because there are no external forces.

Conservation of Momentum : 1 D Steady Inviscid Flow Steady, inviscid 1-D Flow, Body Forces negligible The area of the disturbance is constant.

Conservation of Energy : 1 D Steady Flow Steady flow with negligible Body Forces and no heat transfer in an adiabatic flow For calorically perfect gas: The equation of state for perfect gas reads

Summary of Equations Conservation of mass : Conservation of momentum : Conservation of Energy : The equation of state for perfect gas :

Solution of Simultaneous Equations If the conditions upstream are known, then there are four unknown conditions downstream. A system of four unknowns and four equations is solvable. There exist two solutions because of the quadratic nature of the equations. These two possible solutions may or may not be feasible. Thermodynamics dictates the feasibility of these solutions. Changes in pressure, temperature, volume cannot gaurantee the feasibility of solution. The only tool that helps in identifying the possible solution is change in entropy. For the adiabatic process, the entropy must increase or remain constant.

Machization of Simultaneous Equations : Step 1 Conservation of mass: Dividing this equation by cx

Machization of Momentum Conservation : Step 2 Conservation of momentum: Dividing this through by cx2/g

Merging of Mass & Momentum Conservation Equations thru Machization : Step 3

Machization of Conservation of energy : step 4 Dividing this by

The Unified Equation : Step 5 Simplified Energy Equation : Combined Mass & Momentum Equation : Combined Mass, Momentum and Energy Conservation :

Combined Mass, Momentum and Energy Conservation

Discovery of The Ghost Reversible flow : No entropy generation between x & y,

The Dangerous Ghost Shows existence of an Irreversibility between x & y, The Existence of Normal Shock !!!! This equation relates the downstream Mach number to the upstream Mach Number across a Normal Shock. It can be used to derive pressure ratio, the temperature ratio, and density ratio across a Normal Shock.

Feasibility of Normal Shock The flow across the shock is adiabatic and the stagnation temperature is constant across a shock. The existence of an irreversibile shock will result in a decrease of stagnation pressure. This is feasible iff

Change in entropy for a perfect gas Obtain ratios of thermodynamic parameters across NS. &

An Ultimate Measure for Possibility of a Fantasy Shock is possible only in supersonic flows Infeasible M